Question

select the correct answer from each drop-down menu.in the figure, the ratio of the area of rectangle abef to the area of rectangle acdf is if the coordinates of point a are (0,6), the area of rectangle abef is square units, and the area of rectangle acdf is square units.the perimeter of rectangle bcde is units.reset next

Explanation:

Step1: Find vector $\overrightarrow{AF}$

Point $A(0,6)$, $F(5,2)$
$\overrightarrow{AF} = (5-0, 2-6) = (5, -4)$
Length of $AF$: $\sqrt{5^2 + (-4)^2} = \sqrt{25+16} = \sqrt{41}$

Step2: Find vector $\overrightarrow{AE}$

Point $A(0,6)$, $E(11,10)$
$\overrightarrow{AE} = (11-0, 10-6) = (11, 4)$
Length of $AE$: $\sqrt{11^2 + 4^2} = \sqrt{121+16} = \sqrt{137}$

Step3: Find vector $\overrightarrow{ED}$

Point $E(11,10)$, $D(14,14)$
$\overrightarrow{ED} = (14-11, 14-10) = (3, 4)$
Length of $ED$: $\sqrt{3^2 + 4^2} = \sqrt{9+16} = 5$

Step4: Calculate area of $ABEF$

Area = $|\overrightarrow{AF} \times \overrightarrow{AB}|$, and $\overrightarrow{AB}=\overrightarrow{ED}=(3,4)$
Cross product magnitude: $|5\times4 - (-4)\times3| = |20 +12| = 32$

Step5: Calculate area of $ACDF$

Area = $|\overrightarrow{AF} \times \overrightarrow{AD}|$, $\overrightarrow{AD}=\overrightarrow{AE}+\overrightarrow{ED}=(11+3,4+4)=(14,8)$
Cross product magnitude: $|5\times8 - (-4)\times14| = |40 +56| = 96$

Step6: Find area ratio

Ratio = $\frac{\text{Area }ABEF}{\text{Area }ACDF} = \frac{32}{96} = \frac{1}{3}$

Step7: Calculate perimeter of $BCDE$

$BC=AF=\sqrt{41}$, $ED=5$
Perimeter = $2\times(\sqrt{41} + 5) = 10 + 2\sqrt{41}$

Answer:

1. The ratio of the area of rectangle ABEF to the area of rectangle ACDF is $\frac{1}{3}$
2. The area of rectangle ABEF is 32 square units
3. The area of rectangle ACDF is 96 square units
4. The perimeter of rectangle BCDE is $10 + 2\sqrt{41}$ units